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Abstract

The lamellar grating problem is reformulated through the concept of adaptive spatial resolution. We introduce a new coordinate system such that spatial resolution is increased around the discontinuities of the permittivity function. We derive a new eigenproblem that we solve by using Fourier expansions for both the field and the coefficients of Maxwell’s equations. We provide numerical evidence that highly improved convergence rates can be obtained.

References

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Table 1

Zero and -1-Order Efficiencies and Computation Time for Various Truncation Orders and Two Values of the η Parameter, for the Grating of Figs. 5 and 6

M

η=0.99

η=0

R0

R-1

Time (s)

R0

R-1

Time (s)

TE Polarization

5

0.7288357

0.1404343

0.33

0.8528182

0.1164746(-1)

0.05

6

0.7276024

0.1389257

0.22

0.8197914

0.4044189(-1)

0.06

7

0.7354258

0.1296953

0.33

0.8024120

0.5920369(-1)

0.11

8

0.7344138

0.1317802

0.38

0.7819631

0.8001682(-1)

0.16

9

0.7337542

0.1323167

0.39

0.7725379

0.9018627(-1)

0.17

10

0.7342218

0.1317442

0.49

0.7622694

0.1011264

0.16

11

0.7342712

0.1317277

0.55

0.7571587

0.1066290

0.28

12

0.7342686

0.1317239

0.66

0.7518058

0.1124617

0.27

13

0.7342784

0.1317096

0.71

0.7488667

0.1156276

0.39

14

0.7342788

0.1317094

0.83

0.7458822

0.1189215

0.49

15

0.7342789

0.1317091

0.93

0.7440938

0.1208512

0.50

20

0.7342789

0.1317092

1.6

0.7385729

0.1269484

1

30

0.7342788

0.1317092

4

0.7356116

0.1302258

3

40

0.7342788

0.1317092

9

0.7348521

0.1310702

7

50

0.7342789

0.1317092

15

0.7345754

0.1313783

13

60

0.7342789

0.1317092

28

0.7344516

0.1315164

23

70

0.7342789

0.1317092

45

0.7343881

0.1315872

39

80

0.7342789

0.1317092

67

0.7343523

0.1316272

60

90

0.7342789

0.1317092

96

0.7343306

0.1316514

89

100

0.7342789

0.1317092

127

0.7343166

0.1316670

121

TM Polarization

5

0.8468246

0.1054029

0.33

0.7799993

0.1172988

0.11

6

0.8451876

0.1025578

0.27

0.8348493

0.1059744

0.11

7

0.8489043

0.1038399

0.33

0.8214041

0.1054439

0.11

8

0.8466255

0.1029614

0.33

0.8383793

0.1038340

0.16

9

0.8477851

0.1009658

0.44

0.8280436

0.1044233

0.16

10

0.8483125

0.1017301

0.55

0.8421099

0.1039112

0.22

11

0.8463435

0.1011325

0.55

0.8353895

0.1022652

0.28

12

0.8486235

0.1014897

0.66

0.8395970

0.1032817

0.33

13

0.8475561

0.1015416

0.71

0.8378332

0.1020885

0.44

14

0.8484979

0.1015676

0.88

0.8424096

0.1018666

0.44

15

0.8484465

0.1015990

0.93

0.8418955

0.1016721

0.60

20

0.8484790

0.1015556

1.71

0.8442539

0.1015397

1.10

30

0.8484745

0.1015523

4

0.8457918

0.1015354

3.30

40

0.8484769

0.1015533

8

0.8467749

0.1014714

8

50

0.8484779

0.1015538

16

0.8472149

0.1014855

14

60

0.8484782

0.1015541

27

0.8475163

0.1014872

26

70

0.8484783

0.1015543

43

0.8477069

0.1014946

43

80

0.8484783

0.1015544

70

0.8478444

0.1014992

65

90

0.8484782

0.1015545

98

0.8479448

0.1015036

96

100

0.8484781

0.1015546

138

0.8480210

0.1015070

133

Table 2

Efficiencies for Various Truncation Orders and Three Values of the η Parameter for a Grating That Supports Five Diffraction Ordersa

M

TE Polarization

TM Polarization

R0

R-1

R-2

R0

R-1

R-2

η=0

5

0.4373802

0.3464499 (-1)

0.6587122 (-1)

0.1296643

0.9516329 (-1)

0.6791352 (-2)

7

0.4392082

0.4011332 (-1)

0.8481091 (-1)

0.1424325

0.8943249 (-1)

0.6818339 (-2)

10

0.4453758

0.4276532 (-1)

0.970598 (-1)

0.1710487

0.7882359 (-1)

0.6973433 (-2)

20

0.4464997

0.4397380 (-1)

0.1047476 (-1)

0.1655914

0.8198076 (-1)

0.7547697 (-2)

40

0.4470197

0.4417072 (-1)

0.1061458 (-1)

0.1657705

0.8219983 (-1)

0.7716467 (-2)

60

0.4470891

0.4419122 (-1)

0.1063012 (-1)

0.1659709

0.8219809 (-1)

0.7756200 (-2)

80

0.4471071

0.4419623 (-1)

0.1063310 (-1)

0.1660655

0.8219313 (-1)

0.7771583 (-2)

100

0.4471136

0.4419802 (-1)

0.1063540 (-1)

0.1661172

0.8218982 (-1)

0.7779222 (-2)

η=0.5

5

0.4462756

0.4329902 (-1)

0.9744328 (-2)

0.1464347

0.8920064 (-1)

0.7807272 (-2)

7

0.4454896

0.4404325 (-1)

0.1025768 (-1)

0.1567330

0.8479058 (-1)

0.7528303 (-2)

10

0.4466221

0.4405940 (-1)

0.1052036 (-1)

0.1664648

0.8178478 (-1)

0.7587482 (-2)

20

0.4470335

0.4417579 (-1)

0.1061808 (-1)

0.1658288

0.8219761 (-1)

0.7713203 (-2)

40

0.4471081

0.4419653 (-1)

0.1063422 (-1)

0.1660682

0.8219890 (-1)

0.7770566 (-2)

60

0.4471167

0.4419886 (-1)

0.1063606 (-1)

0.1661492

0.8219141 (-1)

0.7783508 (-2)

80

0.4471190

0.4419945 (-1)

0.1063670 (-1)

0.1661848

0.8218774 (-1)

0.7788435 (-2)

100

0.4471198

0.4419967 (-1)

0.1063663 (-1)

0.1662041

0.8218575 (-1)

0.7790862 (-2)

η=0.95

5

0.4656119

0.4565025 (-1)

0.2732988 (-1)

0.1764076

0.7792956 (-1)

0.8348533 (-2)

7

0.4493744

0.4421632 (-1)

0.1002692 (-1)

0.1650022

0.8304110 (-1)

0.8128166 (-2)

10

0.4471122

0.4420827 (-1)

0.1063671 (-1)

0.1660151

0.8225768 (-1)

0.7731687 (-2)

20

0.4471207

0.4419992 (-1)

0.1063691 (-1)

0.1662376

0.8219317 (-1)

0.7793111 (-2)

40

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662507

0.8218532 (-1)

0.7795947 (-2)

60

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662534

0.8218325 (-1)

0.7796213 (-2)

80

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662546

0.8218242 (-1)

0.7796265 (-2)

100

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662553

0.8218200 (-1)

0.7796277 (-2)

Exact

0.4471416

0.4420008 (-1)

0.1064101 (-1)

0.1659546

0.8219500 (-1)

0.7706766 (-2)

a The grating and incidence parameters are f=0.5θ=0,h=1,d=1,λ=0.365,ν21=1,ν22=1.53-i6.71,ν1=1,ν3=ν22. The assumed exact values are obtained with the classical modal method and a truncation order equal to 61.

Tables (2)

Table 1

Zero and -1-Order Efficiencies and Computation Time for Various Truncation Orders and Two Values of the η Parameter, for the Grating of Figs. 5 and 6

M

η=0.99

η=0

R0

R-1

Time (s)

R0

R-1

Time (s)

TE Polarization

5

0.7288357

0.1404343

0.33

0.8528182

0.1164746(-1)

0.05

6

0.7276024

0.1389257

0.22

0.8197914

0.4044189(-1)

0.06

7

0.7354258

0.1296953

0.33

0.8024120

0.5920369(-1)

0.11

8

0.7344138

0.1317802

0.38

0.7819631

0.8001682(-1)

0.16

9

0.7337542

0.1323167

0.39

0.7725379

0.9018627(-1)

0.17

10

0.7342218

0.1317442

0.49

0.7622694

0.1011264

0.16

11

0.7342712

0.1317277

0.55

0.7571587

0.1066290

0.28

12

0.7342686

0.1317239

0.66

0.7518058

0.1124617

0.27

13

0.7342784

0.1317096

0.71

0.7488667

0.1156276

0.39

14

0.7342788

0.1317094

0.83

0.7458822

0.1189215

0.49

15

0.7342789

0.1317091

0.93

0.7440938

0.1208512

0.50

20

0.7342789

0.1317092

1.6

0.7385729

0.1269484

1

30

0.7342788

0.1317092

4

0.7356116

0.1302258

3

40

0.7342788

0.1317092

9

0.7348521

0.1310702

7

50

0.7342789

0.1317092

15

0.7345754

0.1313783

13

60

0.7342789

0.1317092

28

0.7344516

0.1315164

23

70

0.7342789

0.1317092

45

0.7343881

0.1315872

39

80

0.7342789

0.1317092

67

0.7343523

0.1316272

60

90

0.7342789

0.1317092

96

0.7343306

0.1316514

89

100

0.7342789

0.1317092

127

0.7343166

0.1316670

121

TM Polarization

5

0.8468246

0.1054029

0.33

0.7799993

0.1172988

0.11

6

0.8451876

0.1025578

0.27

0.8348493

0.1059744

0.11

7

0.8489043

0.1038399

0.33

0.8214041

0.1054439

0.11

8

0.8466255

0.1029614

0.33

0.8383793

0.1038340

0.16

9

0.8477851

0.1009658

0.44

0.8280436

0.1044233

0.16

10

0.8483125

0.1017301

0.55

0.8421099

0.1039112

0.22

11

0.8463435

0.1011325

0.55

0.8353895

0.1022652

0.28

12

0.8486235

0.1014897

0.66

0.8395970

0.1032817

0.33

13

0.8475561

0.1015416

0.71

0.8378332

0.1020885

0.44

14

0.8484979

0.1015676

0.88

0.8424096

0.1018666

0.44

15

0.8484465

0.1015990

0.93

0.8418955

0.1016721

0.60

20

0.8484790

0.1015556

1.71

0.8442539

0.1015397

1.10

30

0.8484745

0.1015523

4

0.8457918

0.1015354

3.30

40

0.8484769

0.1015533

8

0.8467749

0.1014714

8

50

0.8484779

0.1015538

16

0.8472149

0.1014855

14

60

0.8484782

0.1015541

27

0.8475163

0.1014872

26

70

0.8484783

0.1015543

43

0.8477069

0.1014946

43

80

0.8484783

0.1015544

70

0.8478444

0.1014992

65

90

0.8484782

0.1015545

98

0.8479448

0.1015036

96

100

0.8484781

0.1015546

138

0.8480210

0.1015070

133

Table 2

Efficiencies for Various Truncation Orders and Three Values of the η Parameter for a Grating That Supports Five Diffraction Ordersa

M

TE Polarization

TM Polarization

R0

R-1

R-2

R0

R-1

R-2

η=0

5

0.4373802

0.3464499 (-1)

0.6587122 (-1)

0.1296643

0.9516329 (-1)

0.6791352 (-2)

7

0.4392082

0.4011332 (-1)

0.8481091 (-1)

0.1424325

0.8943249 (-1)

0.6818339 (-2)

10

0.4453758

0.4276532 (-1)

0.970598 (-1)

0.1710487

0.7882359 (-1)

0.6973433 (-2)

20

0.4464997

0.4397380 (-1)

0.1047476 (-1)

0.1655914

0.8198076 (-1)

0.7547697 (-2)

40

0.4470197

0.4417072 (-1)

0.1061458 (-1)

0.1657705

0.8219983 (-1)

0.7716467 (-2)

60

0.4470891

0.4419122 (-1)

0.1063012 (-1)

0.1659709

0.8219809 (-1)

0.7756200 (-2)

80

0.4471071

0.4419623 (-1)

0.1063310 (-1)

0.1660655

0.8219313 (-1)

0.7771583 (-2)

100

0.4471136

0.4419802 (-1)

0.1063540 (-1)

0.1661172

0.8218982 (-1)

0.7779222 (-2)

η=0.5

5

0.4462756

0.4329902 (-1)

0.9744328 (-2)

0.1464347

0.8920064 (-1)

0.7807272 (-2)

7

0.4454896

0.4404325 (-1)

0.1025768 (-1)

0.1567330

0.8479058 (-1)

0.7528303 (-2)

10

0.4466221

0.4405940 (-1)

0.1052036 (-1)

0.1664648

0.8178478 (-1)

0.7587482 (-2)

20

0.4470335

0.4417579 (-1)

0.1061808 (-1)

0.1658288

0.8219761 (-1)

0.7713203 (-2)

40

0.4471081

0.4419653 (-1)

0.1063422 (-1)

0.1660682

0.8219890 (-1)

0.7770566 (-2)

60

0.4471167

0.4419886 (-1)

0.1063606 (-1)

0.1661492

0.8219141 (-1)

0.7783508 (-2)

80

0.4471190

0.4419945 (-1)

0.1063670 (-1)

0.1661848

0.8218774 (-1)

0.7788435 (-2)

100

0.4471198

0.4419967 (-1)

0.1063663 (-1)

0.1662041

0.8218575 (-1)

0.7790862 (-2)

η=0.95

5

0.4656119

0.4565025 (-1)

0.2732988 (-1)

0.1764076

0.7792956 (-1)

0.8348533 (-2)

7

0.4493744

0.4421632 (-1)

0.1002692 (-1)

0.1650022

0.8304110 (-1)

0.8128166 (-2)

10

0.4471122

0.4420827 (-1)

0.1063671 (-1)

0.1660151

0.8225768 (-1)

0.7731687 (-2)

20

0.4471207

0.4419992 (-1)

0.1063691 (-1)

0.1662376

0.8219317 (-1)

0.7793111 (-2)

40

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662507

0.8218532 (-1)

0.7795947 (-2)

60

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662534

0.8218325 (-1)

0.7796213 (-2)

80

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662546

0.8218242 (-1)

0.7796265 (-2)

100

0.4471206

0.441999 (-1)

0.1063688 (-1)

0.1662553

0.8218200 (-1)

0.7796277 (-2)

Exact

0.4471416

0.4420008 (-1)

0.1064101 (-1)

0.1659546

0.8219500 (-1)

0.7706766 (-2)

a The grating and incidence parameters are f=0.5θ=0,h=1,d=1,λ=0.365,ν21=1,ν22=1.53-i6.71,ν1=1,ν3=ν22. The assumed exact values are obtained with the classical modal method and a truncation order equal to 61.